# Tagged Questions

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### The mth term of a Geometrical Progression is n and nth term is m. Find (m+n)th term

The mth term of a Geometrical Progression is n and nth term is m. Find (m+n)th term. I've tried this: Tm = arm-1 = n (Eq 1) Tn = arn-1 = m (Eq 2) Subracting 2 from 1 rm - r - rn + r = n-m rm - ...
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### How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\$?

How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\$? I thought about it but I could not find a proof. Please give me some hints?
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### Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
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### How is $2\sum_{n=2}^{\infty}\frac{1}{(n-1)(n+1)}=\frac{6}{4}$ calculated?

$$2\sum_{n=2}^{\infty}\frac{1}{(n-1)(n+1)}=\frac{6}{4}$$ I cant figure out why this is $\frac64$. I try to use telescopic series without success.
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### Prove infinite series

$$\frac{1}{x}+\frac{2}{x^2} + \frac{3}{x^3} + \frac{4}{x^4} + \cdots =\frac{x}{(x-1)^2}$$ I can feel it. I can't prove it. I have tested it, and it seems to work. Domain-wise, I think it might be ...
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### Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)}$

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)}$$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
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### Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
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### Confusion Over Sum of Geometric Series

On pg. 88 of A First Course in Probability, it says $$P_i - P_1 = P_1[(q/p) + (q/p)^2 + \cdots + (q/p)^{i-1})]$$ Therefore: $$P_i = \frac{1 - (q/p)^i}{1 - q/p}P_1$$ The series on the right in ...
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### Generalisation of alternating functions

So if we want to have a function go positive negative we take $(-1)^n$, if we want it to take positive positive negative negative(like was on stack exchange a few days ago, we take: ...
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### What is the sum of the power series below?

For $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$ What is the sum of it?
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### How to prove that $\sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$

How to prove $$\sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$$
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### What's wrong with using algebra on infinite series?

I've recently found an article (referred somewhere on this site) criticizing the use of common rules of algebra on infinite series. To be honest, the video referred is one of the videos of Numberphile ...